The hint means, that the "union" of open intervals that are not disjoint is again an open interval. So, if you have any countable union of not necessarily disjoint open intervals, you can make it into a countable union of disjoint open intervals by combining each cluster of intersecting intervals into one interval.

Alternatively you can take for any rational point x in E the "largest" open interval containing x that is still a subset of E.

This is the same as the union of all open intervals containing x, which are subsets of E.

Then show that two such largest intervals must either be disjoint or equal and that their union covers E.